Problem: Simplify; express your answer in exponential form. Assume $t\neq 0, a\neq 0$. $\dfrac{{t^{-4}}}{{(t^{4}a^{-1})^{-5}}}$
To start, try working on the numerator and the denominator independently. In the numerator, we have ${t^{-4}}$ to the exponent ${1}$ . Now ${-4 \times 1 = -4}$ , so ${t^{-4} = t^{-4}}$ In the denominator, we can use the distributive property of exponents. ${(t^{4}a^{-1})^{-5} = (t^{4})^{-5}(a^{-1})^{-5}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{t^{-4}}}{{(t^{4}a^{-1})^{-5}}} = \dfrac{{t^{-4}}}{{t^{-20}a^{5}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{-4}}}{{t^{-20}a^{5}}} = \dfrac{{t^{-4}}}{{t^{-20}}} \cdot \dfrac{{1}}{{a^{5}}} = t^{{-4} - {(-20)}} \cdot a^{- {5}} = t^{16}a^{-5}$.